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Limits and Derivatives
Introduction
Intuitive Idea of Derivatives
Limits
Limits of Trigonometric Functions
Practice: Evaluating Limits
Derivatives
Practice: Derivatives and Differentiation Rules
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Limits and DerivativesIntroduction

Reading time: ~10 min

With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature – WHITEHEAD

On a school trip to a hill station, Aman and Riya lean on the railing of a viewpoint above a deep gorge. Aman lets a small pebble drop. Their physics teacher tells them how far a dropped object falls: about s = 4.9t^2 metres in the first t seconds.

Riya asks a simple-sounding question: how fast is the pebble moving at exactly 2 seconds? Aman is not sure the question even makes sense. A photo taken at exactly 2 seconds shows the pebble frozen in mid-air. In a single instant it covers no distance in no time, and speed is distance divided by time.

Catch the pebble in a speed trap. The green reading is its average speed over a short window just before t = 2, and the purple reading is its average speed over a window of the same length just after. Shrink the window until both readings agree to one decimal place.

Neither reading is the speed at 2 seconds: each is an average over a window. But as the window shrinks, the two averages trap a single number between them, and nothing else fits in the gap. At the instant t = 2 the pebble is falling at m/s.

Chasing averages over smaller and smaller windows, and asking which single value they close in on, is the idea of a limit. The speed it produced is a derivative. Together they are the heart of calculus, the mathematics of how things change. In this chapter we first get a feel for derivatives through the falling pebble, then learn to work with limits properly (including limits of trigonometric functions), and finally define the derivative exactly and find the derivatives of polynomial and trigonometric functions.

A Long Road to Calculus

The question "how fast, at this instant?" is old. In the twelfth century Bhaskara II wrote about tatkalika gati, the "instantaneous motion" of a planet. Two centuries later the Kerala school founded by Madhava found infinite series for the sine and the cosine, ideas that belong to calculus today.

Calculus itself was built in the second half of the seventeenth century by two people working independently: Isaac Newton in England, who needed it for moving planets and falling bodies, and Gottfried Wilhelm Leibniz in Germany, who gave us much of its notation, such as \dfrac{dy}{dx}.

The new methods worked brilliantly, but nobody could say exactly what a limit was. Lagrange introduced the notation f'(x) that we still use. In the 1820s Cauchy took D'Alembert's idea of a limit and made it the foundation: he wrote the ratio \dfrac{\Delta y}{\Delta x} = \dfrac{f(x + i) - f(x)}{i} and defined the derivative as its limit as i \to 0. Karl Weierstrass later made the definition of a limit completely precise.

For a long time calculus was thought too hard for young students. Around 1900, John Perry and others in England argued that its main ideas are simple and belong in school, and F. L. Griffin began teaching calculus to first-year college students, a daring step at the time. Today physics, chemistry, economics and biology all run on calculus.

Drag these milestones into the order in which they happened, earliest at the top.

Lagrange writes the derivative as f′(x)
Bhaskara II writes about the instantaneous motion of a planet
John Perry argues that calculus belongs in schools
Newton and Leibniz, independently, build calculus
Cauchy defines the derivative as a limit
Madhava's Kerala school finds infinite series for sine and cosine

Eight centuries separate Bhaskara's moving planets from your falling pebble, and the question has not changed: how fast, at this very instant? Let us start answering it.

Sina